**Abstract: **
I will answer the following problem: Let I be a differential invariant, no
matter how it was computed,
determine systematically the corresponding approximating discrete invariant.
Our approach takes
advantage of the new geometrical construction of multi-spaces introduced by
P. Olver where we can
switch at our convenience from/to normal derivatives to/from divided
differences. Multi-spaces are
a far reaching generalization of the classical blow-up construction in
algebraic geometry. New
examples that heretofore were intractable are now easily solved.